We develop a constructive method to derive exactly solvable quantum
mechanical models of rational (Calogero) and trigonometric (Sutherland) type.
This method starts from a linear algebra problem: finding eigenvectors of
triangular finite matrices. These eigenvectors are transcribed into
eigenfunctions of a selfadjoint Schr\"odinger operator. We prove the
feasibility of our method by constructing a new "AG3​ model" of trigonometric
type (the rational case was known before from Wolfes 1975). Applying a Coxeter
group analysis we prove its equivalence with the B3​ model. In order to
better understand features of our construction we exhibit the F4​ rational
model with our method.Comment: 22 pages, 2 eps figures, latex 2epsilon, usepackage epsfi